A145439 -id:A145439 - cp's OEIS frontend

A157697 Decimal expansion of sqrt(2/3).

8, 1, 6, 4, 9, 6, 5, 8, 0, 9, 2, 7, 7, 2, 6, 0, 3, 2, 7, 3, 2, 4, 2, 8, 0, 2, 4, 9, 0, 1, 9, 6, 3, 7, 9, 7, 3, 2, 1, 9, 8, 2, 4, 9, 3, 5, 5, 2, 2, 2, 3, 3, 7, 6, 1, 4, 4, 2, 3, 0, 8, 5, 5, 7, 5, 0, 3, 2, 0, 1, 2, 5, 8, 1, 9, 1, 0, 5, 0, 0, 8, 8, 4, 6, 6, 1, 9, 8, 1, 1, 0, 3, 4, 8, 8, 0, 0, 7, 8, 2, 7, 2, 8, 6, 4

Offset: 0

R. J. Mathar, Mar 04 2009

Height (from a vertex to the opposite face) of regular tetrahedron with unit edge. - Stanislav Sykora, May 31 2012
The eccentricity of the ellipse of minimum area that is circumscribing two equal and externally tangent circles (Kotani, 1995). - Amiram Eldar, Mar 06 2022
The standard deviation of a roll of a 3-sided die. - Mohammed Yaseen, Feb 23 2023

			0.81649658092772603273242802490196379732198249355222...

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (168) on page 32.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Jisho Kotani, Problem 2053, Crux Mathematicorum, Vol. 5, No. 1 (1995), p. 202; Solution to Problem 2053, by David Hankin, ibid., Vol. 22, No. 4 (1996), pp. 187-188.
D. H. Lehmer, Interesting series involving the Central Binomial Coefficient, Am. Math. Monthly, Vol. 92, No. 7 (1985), pp. 449-457.
Index entries for algebraic numbers, degree 2.

Cf. A002193, A020763, A092259, A115754, A145439.

Sqrt(2/3); // G. C. Greubel, Mar 30 2018

Maple
```
evalf(sqrt(2/3)) ;
```

RealDigits[Sqrt[2/3], 10, 200][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011*)

sqrt(2/3) \\ G. C. Greubel, Mar 30 2018

Equals 1 - (1/2)/2 + (1*3)/(2*4)/2^2 - (1*3*5)/(2*4*6)/2^3 + ... [Jolley]
Equals Sum_{n>=0} (-1)^n*binomial(2n,n)/8^n = 1/A115754. Averaging this constant with sqrt(2) = A002193 = Sum_{n>=0} binomial(2n,n)/8^n yields A145439.
From Michal Paulovic, Dec 08 2022: (Start)
Equals 2 * A020763.
Has periodic continued fraction expansion [0, 1, 4; 2, 4]. (End)
Equals exp(-arctanh(1/5)). - Amiram Eldar, Jul 10 2023
Equals Product_{k>=1} (1 + (-1)^k/A092259(k)). - Amiram Eldar, Nov 24 2024

A092259 Numbers that are congruent to {4, 8} mod 12.

Original entry on oeis.org

4, 8, 16, 20, 28, 32, 40, 44, 52, 56, 64, 68, 76, 80, 88, 92, 100, 104, 112, 116, 124, 128, 136, 140, 148, 152, 160, 164, 172, 176, 184, 188, 196, 200, 208, 212, 220, 224, 232, 236, 244, 248, 256, 260, 268, 272, 280, 284, 292, 296, 304, 308, 316, 320, 328, 332

Offset: 1

Giovanni Teofilatto, Feb 19 2004

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001651, A017617, A017569, A069587, A092899, A145439, A157697.

Fourth row of A092260.

[n : n in [0..400] | n mod 12 in [4, 8]]; // Wesley Ivan Hurt, May 21 2016

A092259:=n->6*n-3-(-1)^n: seq(A092259(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Table[6n-3-(-1)^n, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)
LinearRecurrence[{1,1,-1},{4,8,16},60] (* Harvey P. Dale, Oct 07 2021 *)

G.f.: 4*x*(1+x+x^2) / ( (1+x)*(x-1)^2 ).
a(n) = 4 * A001651(n).
Iff phi(n) = phi(3n/2), then n is in A069587. - Labos Elemer, Feb 25 2004
a(n) = 12*(n-1)-a(n-1) (with a(1)=4). - Vincenzo Librandi, Nov 16 2010
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
a(n) = 6n - 3 - (-1)^n.
a(2n) = A017617(n-1) for n>1, a(2n-1) = A017569(n-1) for n>1.
a(n) = -a(1-n), a(n) = A092899(n) + 1 for n>0. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi*sqrt(3)/36. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 24 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 1/sqrt(2) + 1/sqrt(6) (A145439).
Product_{n>=1} (1 + (-1)^n/a(n)) = sqrt(2/3) (A157697). (End)

Edited and extended by Ray Chandler, Feb 21 2004

A127692 Expansion of psi(x^4) + x * psi(x^12) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Jan 19 2007

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

a(n) = 1 if n is four times a triangular number or one more than twelve times a triangular number else 0. - Michael Somos, Jul 19 2012

			G.f. = 1 + x + x^4 + x^12 + x^13 + x^24 + x^37 + x^40 + x^60 + x^73 + x^84 + ...
G.f. = q + q^3 + q^9 + q^25 + q^27 + q^49 + q^75 + q^81 + q^121 + q^147 + q^169 + ...

Antti Karttunen, Table of n, a(n) for n = 0..10000
Richard Blecksmith, John Brillhart, and Irving Gerst, Some infinite product identities, Math. Comp. 51 (1988), no. 183, 301-314. MR0942157 (89f:05017)
Shaun Cooper and Michael Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001), 131-139. see p. 134 Theorem 5.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A005369, A010054, A080995, A089806, A145439.

Cf. A000122, A000700, A010054, A121373.

{a(n) = issquare(2*n + 1) + issquare(6*n + 3)};

{a(n) = n = 2*n + 1; issquare(n) || issquare(3*n)};

Euler transform of period 24 sequence [ 1, -1, 0, 1, -1, 1, -1, 0, 0, 0, 1, -1, 1, 0, 0, 0, -1, 1, -1, 1, 0, -1, 1, -1, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 1, else b(p^e) = (1 + (-1)^e)/2.
a(3*n + 1) = a(n), a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = a(6*n + 3) = 0.
a(2*n) = A005369(n). a(4*n) = A010054(n). a(6*n) = A089806(n). a(12*n) = A080995(n).
G.f.: Sum_{k>0} x^(2k(k-1)) +x^(6k(k-1)+1) = Product_{k>0} (1-x^(24k)) (1-x^(24k-5)) (1-x^(24k-7)) (1-x^(24k-17)) (1-x^(24k-19)) (1+x^(12k-1)) (1+x^(12k-4)) (1+x^(12k-6)) (1+x^(12k-8)) (1+x^(12k-11)).
From Michael Somos, Jul 19 2012: (Start)
Expansion of f(x, -x^5) * f(-x^4, -x^8) / f(x, -x) in powers of x where f(,) is the Ramanujan two-variable theta function.
G.f.: (Sum_{k in Z} x^(2*k*(k + 1)) + x^(6*k*(k + 1) + 1)) / 2.
a(n) = A195198(2*n + 1). (End)
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 1/sqrt(2) + 1/sqrt(6) = 1.115355... (A145439). - Amiram Eldar, Dec 29 2023

A337402 Decimal expansion of the length of third shortest diagonal in a regular 12-gon with unit edge length.

Original entry on oeis.org

3, 3, 4, 6, 0, 6, 5, 2, 1, 4, 9, 5, 1, 2, 3, 1, 6, 2, 2, 3, 0, 1, 1, 7, 5, 1, 2, 3, 6, 6, 7, 4, 9, 2, 8, 1, 3, 8, 3, 7, 4, 8, 1, 5, 5, 3, 3, 9, 3, 7, 5, 7, 1, 7, 3, 9, 8, 1, 3, 6, 5, 8, 9, 0, 6, 1, 1, 5, 7, 8, 9, 0, 6, 4, 2, 1, 8, 1, 8, 0, 7, 1, 5, 4, 5, 5, 1

Offset: 1

Mohammed Yaseen, Aug 26 2020

The distinct diagonal lengths in a regular 12-gon ABC...JKL with unit edge length are
AC = sqrt(2 +   sqrt(3)) =     sqrt(2)/(-1+sqrt(3)) = A188887
AD = sqrt(4 + 2*sqrt(3)) =          2 /(-1+sqrt(3)) = A090388
AE = sqrt(6 + 3*sqrt(3)) =     sqrt(6)/(-1+sqrt(3))
AF = sqrt(7 + 4*sqrt(3)) = (1+sqrt(3))/(-1+sqrt(3)) = A019973
AG = sqrt(8 + 4*sqrt(3)) =   2*sqrt(2)/(-1+sqrt(3)) = A214726

			3.34606521495123162230117512366749281383748155339375...

Paolo Xausa, Table of n, a(n) for n = 1..10000
N. J. A. Sloane and Gavin A. Theobald, On Dissecting Polygons into Rectangles, arXiv:2309.14866 [math.CO], 2023. See Eq. (2.3).
I. J. Zucker, G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Cam. Phil. Soc. 131 (2001) 309 eq (8.9)

Cf. A337301, A188887, A090388, A019973, A214726, A145439.

First[RealDigits[Sqrt[6+3Sqrt[3]],10,100]] (* Paolo Xausa, Oct 19 2023 *)

sqrt(6 + 3*sqrt(3)) \\ Michel Marcus, Aug 26 2020

Equals sin(Pi/3)/sin(Pi/12) = sqrt(2) + 2*cos(Pi/12) = sqrt(3*cot(Pi/12)).
Equals sqrt(6 + 3*sqrt(3)) = sqrt(6)/(-1+sqrt(3)) = (3+sqrt(3))/sqrt(2).
Equals 3*A145439.
Equals Gamma(1/24)*Gamma(11/24)/(Gamma(5/24)*Gamma(7/24)) [Zucker] - R. J. Mathar, Jun 24 2024

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A157697 Decimal expansion of sqrt(2/3).

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A092259 Numbers that are congruent to {4, 8} mod 12.

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A127692 Expansion of psi(x^4) + x * psi(x^12) in powers of x where psi() is a Ramanujan theta function.

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A337402 Decimal expansion of the length of third shortest diagonal in a regular 12-gon with unit edge length.

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